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We define and analyze the notion of variational Wardrop equilibrium for nonatomic aggregative games with an infinity of players types. These equilibria are characterized through an infinite-dimensional variational inequality. We show, under monotonicity conditions, a convergence theorem enables to approximate such an equilibrium with arbitrary precision. To this end, we introduce a sequence of nonatomic games with a finite number of players types, which approximates the initial game. We show the existence of a symmetric Wardrop equilibrium in each of these games. We prove that those symmetric equilibria converge to an equilibrium of the infinite game, and that they can be computed as solutions of finite-dimensional variational inequalities. The model is illustrated through an example from smart grids: the description of a large population of electricity consumers by a parametric distribution gives a nonatomic game with an infinity of different players types, with actions subject to coupling constraints., arXiv admin note: substantial text overlap with arXiv:1806.06230

After defining a pure-action profile in a nonatomic aggregative game, where players have specific compact convex pure-action sets and nonsmooth convex cost functions, as a square-integrable function, we characterize a Wardrop equilibrium as a solution to an infinite-dimensional generalized variational inequality. We show the existence of Wardrop equilibrium and variational Wardrop equilibrium, a concept of equilibrium adapted to the presence of coupling constraints, in monotone nonatomic aggregative games. The uniqueness of (variational) Wardrop equilibrium is proved for strictly or aggregatively strictly monotone nonatomic aggregative games. We then show that, for a sequence of finite-player aggregative games with aggregative constraints, if the players' pure-action sets converge to those of a strongly (resp. aggregatively strongly) monotone nonatomic aggregative game, and the aggregative constraints in the finite-player games converge to the aggregative constraint of the nonatomic game, then a sequence of so-called variational Nash equilibria in these finite-player games converge to the variational Wardrop equilibrium in pure-action profile (resp. aggregate-action profile). In particular, it allows the construction of an auxiliary sequence of games with finite-dimensional equilibria to approximate the infinite-dimensional equilibrium in such a nonatomic game. Finally, we show how to construct auxiliary finite-player games for two general classes of nonatomic games., 23 pages, 2 figures